The boy’s hat is under his arm.  In this expression, what relation does the preposition under show?  You know that hat and arm are words used as signs of two objects, or ideas; but under is not the sign of a thing you can think of:  it is merely the sign of the relation existing between the two objects.  Hence you may perceive, that since the word under is the sign of the relation existing between particular ideas, it also expresses a relation existing between the words hat and arm, which words are the representatives of those ideas.

The boy holds his hat in his hand.  In this sentence the preposition in shows the relation existing between hat and hand, or the situation, or relative position, each has in regard to the other.  And, if I say, The boy’s hat is on his head, you perceive that on shows the relation between hat and head.  Again, in the expressions, The boy threw his hat up stairs—­under the bed—­behind the table—­through the window—­over the house—­across the street—­into the water—­and so on, you perceive that the several prepositions express the different relations existing between the hat and the other nouns, stairs, bed, table, window, house, street, and water.

A preposition tells where a thing is:  thus, “The pear is on the ground, under the tree.”

Prepositions govern the objective case, but they do not express an action done to some object, as an active-transitive verb or participle does.  When a noun or pronoun follows a preposition, it is in the objective case, because it is the object of the relation expressed by the preposition, and not the object of an action.

I can now give you a more extensive explanation of the objective case, than that which was given in a former lecture.  I have already informed you, that the objective case expresses the object of an action or of a relation; and, also, that there are three parts of speech which govern nouns and pronouns in the objective case, namely, active-transitive verbs, participles derived from transitive verbs, and prepositions.  A noun or pronoun in the objective case, cannot be, at the same time, the object of an action and of a relation.  It must be either the object of an action or of a relation.  And I wish you particularly to remember, that whenever a noun or pronoun is governed by a transitive verb or participle, it is the object of an action; as, The tutor instructs his pupils; or, The tutor is instructing his pupils; but whenever a noun or pronoun is governed by a preposition, it is the object of a relation; as, The tutor gives good instruction to his pupils.